GCD-only computation of the oy-splitting factorization

Determine whether the oy-splitting factorization p = ps pn of a polynomial p in F[y]—with ps being oy-special and pn having only oy-normal irreducible factors—can be computed using greatest common divisor computations alone (without full factorization), analogous to the gcd-based approach available in the differential case.

Background

The authors introduce oy-normal and oy-special polynomials and define the oy-splitting factorization p = ps pn, which separates the special and normal components. This factorization underpins their canonical representation of rational functions and subsequent reduction algorithms.

In the differential setting, analogous splitting can be obtained using gcd computations without full factorization. Establishing whether a similar gcd-only procedure exists in the (q-)difference context would improve efficiency and practicality of reductions and canonical forms for (q-)hypergeometric terms.